Decomposition numbers for abelian defect RoCK blocks of double covers of symmetric groups

Abstract

We calculate the (super)decomposition matrix for a RoCK block of a double cover of the symmetric group with abelian defect, verifying a conjecture of the first author. To do this, we exploit a theorem of the second author and Livesey that a RoCK block (Formula presented.) is Morita superequivalent to a wreath superproduct of a certain quiver (super)algebra with the symmetric group (Formula presented.). We develop the representation theory of this wreath superproduct to compute its Cartan invariants. We then directly construct projective characters for (Formula presented.) to calculate its decomposition matrix up to a triangular adjustment, and show that this adjustment is trivial by comparing Cartan invariants.